First of all, I don't know who was responsible for puppeteering Ernie's feet when he's on the swing, but whoever it is deserves special recognition for brilliant subtlety. I know intellectually that Ernie doesn't usually have feet, but I have a hard time believing it because whenever he needs to have feet, somehow they always appear, so I tend to just take for granted that they're there all the time.
This song is a Sesame Street classic, and it's been reinterpreted many, many times, most recently during the Macy's Thanksgiving Day parade. It was written by Joe Raposo, who was the original Sesame Street music director and who wrote (and often performed) many, many, many songs for the show, including the theme song, which I mention because the Sesame Street theme and "Somebody Come and Play" seem to be structurally related. I don't know if that was intentional.
Raposo was a fascinating and complicated personality. I could write an entire essay on him and his contrasting faults and virtues, and maybe I will someday if I have the time and inclination. By all accounts, he was a musical genius, a versatile virtuoso who thrived on spontaneity. He made a very deliberate effort to give Sesame Street a sophisticated musical sound (and God bless him for it, because I give a LOT of credit to Sesame Street for informing my eclectic musical taste). I think "Somebody Come and Play," is an excellent example of that sophistication: it is deceptively upbeat and cheery-sounding, in contrast with the lyrics, which are wistful and even a little melancholy. The more you listen to it and the more you think about it, the more depth you uncover in it. Even today, that goes against the conventional wisdom that children's music (and children's entertainment in general) has to be simplistic, so imagine for a moment how radical it must have seemed in 1969.